Course Projects: Simulation, PDEs, Causal Inference, and LLMs
This repository collects several projects from my coursework in mathematics and statistics at the University of Michigan, Ann Arbor.
1. Monte Carlo Simulation of the St. Petersburg Game
This is a coding project for STATS 506, where I implement simulations of the St. Petersburg game and document the functions as a custom R package.
The St. Petersburg game is a classical example where the payoff
\[ X = 2^K, \qquad \mathbb{P}(X = 2^k) = 2^{-k}, \quad k \ge 1 \]
has infinite expectation, yet empirical averages behave very irregularly. Monte Carlo simulations of repeated plays allow us to visualize and study the convergence behavior of the sample mean and scaled averages
\[ A_n = \frac{1}{n \log_2 n} \sum_{i=1}^n X_i . \]
We also discuss the connection of this problem to robust estimation.
2. PDE Simulation with PINNs and Classical Numerical Methods
This project is related to coursework in MATH 440, MATH 656, and MATH 651. It focuses on time-dependent PDEs (e.g., the 1D Saint-Venant / shallow-water system) and compares:
Physics-Informed Neural Networks (PINNs)
PINNs approximate the solution by minimizing PDE residuals via automatic differentiation.
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Figure 1. Solutions of the 1D Saint-Venant equations using PINNs.
Classical Numerical Methods
Classical numerical methods such as finite-volume schemes provide high-resolution reference solutions.
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